Bridge pins are little but they can can be really difficult to get out of their holes.
Here are couple of safe and easy ways to do the job.

The first thing you should try is to simply push the string back down the hole a
bit:
It's a simple thing, and it can often release the wedged string and bridge pin.

Using the same diagonal cutters you use to trim the loose end of the strings, just
close the jaws lightly under the head of the bridge pin:

Then, lifting rather that prying, just pull the bridge pin straight up and out:

Most of the time that's all it takes.

If the pin is too tight to lift, you can protect the top of the bridge with a little
slip of cardboard, or better yet, a scrap of hardwood veneer, and pry the pin right
out:
Prying this way gives you terrific leverage, and it's possible to break the pin in
half, so it pays to be careful. If it seems likely to break, try pushing the string
and pin back down together to release the string, then pry again.

Even if you pull really hard, you won't scar the head of the pin because you're gripping
it right in the turned shoulder where little nicks won't show:

Some of those little plastic string winding cranks come with special notches that
allow you to pry out stubborn bridge pins:
While they don't work in as many different situations as the diagonal cutters, these
plastic pry devices do afford a better sense of safety. . .

Another, even safer way to release a bridge pin is to slack the strings, and reach
inside the guitar with a coin, simply pushing out the pin from the inside: